Karl Vind
Springer Berlin Heidelberg
2003e druk, 2002
9783540416838
Independence, Additivity, Uncertainty
Specificaties
Gebonden, 277 blz.
|
Engels
Springer Berlin Heidelberg |
2003e druk, 2002
ISBN13: 9783540416838
Rubricering
Onderdeel van serie
Studies in Economic Theory
Levertijd ongeveer 9 werkdagen
Gratis verzonden
Samenvatting
Volume 14 in the Studies in Economic Theory series deals with the important economic problem of uncertainty. It contains all the classical results, but also new results that give a solution to how uncertainty can be formalized.
Specificaties
ISBN13:9783540416838
Taal:Engels
Bindwijze:gebonden
Aantal pagina's:277
Uitgever:Springer Berlin Heidelberg
Druk:2003
Hoofdrubriek:Economie
Inhoudsopgave
1 Introduction.- 1.1 Economics.- 1.2 Statistics.- 1.3 Mathematic.- 1.4 Summary of results.- 1.5 Applications.- I Basic Mathematics.- 2 Totally preordered sets.- 2.1 Introduction.- 2.2 Order relations.- 2.2.1 Basic concepts.- 2.2.2 Completion.- 2.2.3 Representation.- 2.3 Topological concepts.- 2.4 The order topology.- 2.5 Representation.- 2.6 Notes.- 2.6.1 Basic concepts.- 2.6.2 Ordered sets.- 2.6.3 Topology and order topology.- 2.6.4 Ordered topological spaces.- 2.6.5 Lexicographic orders.- 2.6.6 Removing gaps.- 2.6.7 Further results.- 3 Preferences and preference functions.- 3.1 Introduction.- 3.2 Representations and representation theorems..- 3.3 Notes.- 4 Totally preordered product sets.- 4.1 Introduction.- 4.2 Independence assumptions.- 4.3 Order topologies on product sets.- 4.4 Existence of real continuous order homomorphisms.- 4.5 Note.- 5 A subset of a product set.- 5.1 Introduction.- 5.2 Independence.- 5.3 A total preorder on the set SA.- 5.4 The Thomsen and the Reidemeister conditions.- 5.5 Note.- 5.5.1 The Reidemeister and Thomsen conditions.- 6 Mean groupoids.- 6.1 Introduction.- 6.2 Definition of a commutative mean groupoid.- 6.3 Completion of commutative mean groupoid.- 6.4 The Aczél Fuchs theorem.- 6.5 Extension of a commutative mean groupoid.- 6.6 The bisymmetry equation.- 6.7 Notes.- 6.7.1 History and other results.- 6.7.2 Classifying commutative mean groupoids.- 6.7.3 Lexicographic “mean groupoids”.- 6.7.4 Totally ordered mixture spaces.- 6.7.5 Reducible.- 6.7.6 Products of mean groupoids.- 6.7.7 Completion.- 6.7.8 Measurement of magnitudes.- 6.7.9 The bisymmetry equation.- 6.7.10 Counter example (Andrew Gleason, Harvard).- 7 Products of two sets as a mean groupoid.- 7.1 Introduction.- 7.2 Thomsen’s and Reidemeister’s conditions.- 7.3 (S, % MathType!MTEF!2!1!+-
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\kern-\nulldelimiterspace} {{ \sim _x},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{o_x}{\square _x}}}} \right)$$.- 9.8 Notes.- III Relations on Measures.- 10 Relations on sets of probability measures.- 10.1 Introduction.- 10.2 Definitions and mathematics.- 10.3 Existence of a Bernoulli function.- 10.4 von Neumann Morgenstern preferences.- 10.4.1 The finite case.- 10.4.2 The general case.- 10.4.3 Special cases.- 10.5 Notes.- IV Integral Representations.- 11 A general integral representation by Birgit Grodal.- 11.1 Introduction.- 11.2 Existence of u : X × Y ? ? with % MathType!MTEF!2!1!+-
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% gUIiYdGccaWGKbGaamyDaaaa!482F!$$f\left( {g,A} \right) = \int_A {u\left( {x,g\left( x \right)} \right)} d\mu $$.- 11.3 Continuity and boundedness of u.- 11.4 Existence of u: X × Y ? ? when G is a set of measurable selections..- 11.5 Notes.- 12 Special integral representations by Birgit Grodal.- 12.1 Introduction.- 12.2 % MathType!MTEF!2!1!+-
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% niabgUIiYdGccaWGKbGaeq4UdWgaaa!4972!$$f\left( {g,A} \right) = \int_A {\bar u\left( t \right){e^{ - \delta t}}} d\lambda $$.- 12.5 Notes.- V Decompositions and Uncertainty.- 13 Decompositions. Uncertainty.- 13.1 Introduction.- 13.2 von Neumann Morgenstern preferences.- 13.3 Function spaces.- 13.3.1 Y = Z = {0, 1}. Subjective probabilities and uncertainty.- 13.3.2 X = {1, 2, ... , n}(?i?XYi, P).- 13.3.3 Y and X general.- 13.4 Historical notes.- 13.4.1 Knight.- 13.4.2 Keynes.- 13.4.3 von Neumann Morgenstern.- 13.4.4 Savage.- 13.4.5 Aumann.- 13.4.6 Friedman.- 13.4.7 Bewley.- 13.5 Conclusion.- 14 Uncertainty on products.- 14.1 Introduction.- 14.2 One level uncertainty on factors and products.- 14.2.1 Y = Z = {0, 1}, X = X1 × X2.- 14.2.2 Y = Z = {0, 1}, (X, Ai)i?I.- 14.2.3 Y and Z general.- 14.3 Two level uncertainty.- 14.4 Conclusions.- 14.5 Note.- 15 Conditional uncertainty.- 15.1 Introduction.- 15.2 Relations on function spaces.- 15.3 One probability-uncertainty measure.- 15.4 Several probability-uncertainty measures.- 15.4.1 Y = Z = {0, 1}.- 15.4.2 Y and Z general.- 15.5 Two level uncertainty.- 15.6 Conclusion.- VI Applications.- 16 Production, utility, preference.- 16.1 Introduction.- 16.2 Production functions.- 16.3 Additive preference functions.- 16.4 Additive utility functions.- 16.5 Notes.- 17 Preferences over time.- 17.1 Introduction.- 17.2 ((T, A), Y, Z, G, H, P) Existence of f : G × H × A ? ?.- 17.2.1 Y general.- 17.3 Existence and decomposition of f : G × H × G × H × A ? ?.- 17.4 Notes.- 18 A foundation for statistics.- 18.1 Introduction and historical background.- 18.2 Basic concepts.- 18.3 Uncertainty about the parameter space..- 18.4 Robust Bayesian inference.- 18.5 Requirements for a foundation of statistics..- 18.6 A foundation of statistics.- 18.7 Notes.- References.
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% baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$).- 8.9 Y a commutative mean groupoid.- 8.9.1 (H, % MathType!MTEF!2!1!+-
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% baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$, o).- 8.10 Y a commutative mean groupoid with zero.- 8.10.1 (H, % MathType!MTEF!2!1!+-
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% baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$ , ox, ?x).- 8.11 Related functional equations.- 8.12 Notes.- 9 Relations on function spaces.- 9.1 Introduction.- 9.2 Existence of F : G ? ?, f : G × A ? ?.- 9.3 Existence of F : G × H ? ?, f : G × H × A ? ?.- 9.3.1 ((X, A), Y, G, P) Existence of F : G × G ? ? , f : G × G × A ? ?.- 9.4 X = {1, 2, ... , n} (?i?XYi, ?i?XZi, P).- 9.5 Y = Z = {0,1}, (X, A, P).- 9.6 Minimal independence assumptions.- 9.7 (Yx, Qx)x?X.- 9.7.1 (X, Y, G, Q, (Qx)x?X.- 9.7.2 ((G, % MathType!MTEF!2!1!+-
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% baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$ o), (Yx, Px)x?X) = ((G, % MathType!MTEF!2!1!+-
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% baaaa!37AD!$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \succ } $$, o), (Yx × Yx) / ~x, % MathType!MTEF!2!1!+-
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% badaWgaaWcbaGaamiEaaqabaaaaa!38D5!$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } _x}$$, ox).- 9.7.3 (G, P),(Yx/ ~x, % MathType!MTEF!2!1!+-
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% badaWgaaWcbaGaamiEaaqabaaaaa!38D5!$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } _x}$$, ox).- 9.7.4 % MathType!MTEF!2!1!+-
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% beaaaOGaayjkaiaawMcaaaaa!4DEF!$$\left( {G,{\text{ P}}} \right),{\text{ }}\left( {{{{Y_x}} \mathord{\left/
{\vphantom {{{Y_x}} {{ \sim _x},{\text{ }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\text{ }}{{\text{o}}_x}}}} \right.
\kern-\nulldelimiterspace} {{ \sim _x},{\text{ }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{\text{ }}{{\text{o}}_x}}}{\square _x}} \right)$$.- 9.7.5 % MathType!MTEF!2!1!+-
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% aaaa!65EF!$$\left( {\left( {G,P} \right),{{\left( {{Y_x},{P_x}} \right)}_{x \in X}}} \right) = \left( {{{\left( {G \times G,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } ,o,\square } \right),\left( {Y \times Y} \right)} \mathord{\left/ {\vphantom {{\left( {G \times G,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } ,o,\square } \right),\left( {Y \times Y} \right)} {{ \sim _x},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{o_x}{\square _x}}}} \right.
\kern-\nulldelimiterspace} {{ \sim _x},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } }_x},{o_x}{\square _x}}}} \right)$$.- 9.8 Notes.- III Relations on Measures.- 10 Relations on sets of probability measures.- 10.1 Introduction.- 10.2 Definitions and mathematics.- 10.3 Existence of a Bernoulli function.- 10.4 von Neumann Morgenstern preferences.- 10.4.1 The finite case.- 10.4.2 The general case.- 10.4.3 Special cases.- 10.5 Notes.- IV Integral Representations.- 11 A general integral representation by Birgit Grodal.- 11.1 Introduction.- 11.2 Existence of u : X × Y ? ? with % MathType!MTEF!2!1!+-
% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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% gUIiYdGccaWGKbGaamyDaaaa!482F!$$f\left( {g,A} \right) = \int_A {u\left( {x,g\left( x \right)} \right)} d\mu $$.- 11.3 Continuity and boundedness of u.- 11.4 Existence of u: X × Y ? ? when G is a set of measurable selections..- 11.5 Notes.- 12 Special integral representations by Birgit Grodal.- 12.1 Introduction.- 12.2 % MathType!MTEF!2!1!+-
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% abeY7aTbaa!4C2D!$$f\left( {g,A} \right) = \int_A {\beta \left( {x,\bar u\left( {g\left( x \right)} \right)} \right)} d\mu $$.- 12.3 % MathType!MTEF!2!1!+-
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% gaaa!4B7B!$$f\left( {g,A} \right) = \int_A {\bar u} \left( {g\left( x \right)} \right)\alpha \left( x \right)d\mu $$.- 12.4 % MathType!MTEF!2!1!+-
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% niabgUIiYdGccaWGKbGaeq4UdWgaaa!4972!$$f\left( {g,A} \right) = \int_A {\bar u\left( t \right){e^{ - \delta t}}} d\lambda $$.- 12.5 Notes.- V Decompositions and Uncertainty.- 13 Decompositions. Uncertainty.- 13.1 Introduction.- 13.2 von Neumann Morgenstern preferences.- 13.3 Function spaces.- 13.3.1 Y = Z = {0, 1}. Subjective probabilities and uncertainty.- 13.3.2 X = {1, 2, ... , n}(?i?XYi, P).- 13.3.3 Y and X general.- 13.4 Historical notes.- 13.4.1 Knight.- 13.4.2 Keynes.- 13.4.3 von Neumann Morgenstern.- 13.4.4 Savage.- 13.4.5 Aumann.- 13.4.6 Friedman.- 13.4.7 Bewley.- 13.5 Conclusion.- 14 Uncertainty on products.- 14.1 Introduction.- 14.2 One level uncertainty on factors and products.- 14.2.1 Y = Z = {0, 1}, X = X1 × X2.- 14.2.2 Y = Z = {0, 1}, (X, Ai)i?I.- 14.2.3 Y and Z general.- 14.3 Two level uncertainty.- 14.4 Conclusions.- 14.5 Note.- 15 Conditional uncertainty.- 15.1 Introduction.- 15.2 Relations on function spaces.- 15.3 One probability-uncertainty measure.- 15.4 Several probability-uncertainty measures.- 15.4.1 Y = Z = {0, 1}.- 15.4.2 Y and Z general.- 15.5 Two level uncertainty.- 15.6 Conclusion.- VI Applications.- 16 Production, utility, preference.- 16.1 Introduction.- 16.2 Production functions.- 16.3 Additive preference functions.- 16.4 Additive utility functions.- 16.5 Notes.- 17 Preferences over time.- 17.1 Introduction.- 17.2 ((T, A), Y, Z, G, H, P) Existence of f : G × H × A ? ?.- 17.2.1 Y general.- 17.3 Existence and decomposition of f : G × H × G × H × A ? ?.- 17.4 Notes.- 18 A foundation for statistics.- 18.1 Introduction and historical background.- 18.2 Basic concepts.- 18.3 Uncertainty about the parameter space..- 18.4 Robust Bayesian inference.- 18.5 Requirements for a foundation of statistics..- 18.6 A foundation of statistics.- 18.7 Notes.- References.
